Performance analysis of space time block coded systems with channel estimation

The resultsshow that the selection combining systems achieve the diversity gain provided by thetotal number of available receive antennas, but independent of the number of antennaschosen

Space-Time Block Coded Systems

with Channel Estimation

Shan Cheng

M.Eng, Zhejiang University, P.R China

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

May 2006

I would like to express my profound gratitude to my supervisors: Dr A lanathan and Prof P Y Kam, for their invaluable guidance and endless patiencethroughout the entire duration of my Ph.D course

Nal-I would like to thank my parents and other family members Their love, patienceand understanding have accompanied me all the way along Special regards to mybeloved grandfather, who departed us in 2004

I am also thankful to my labmates and friends, not only for their resourceful cussion in research, but their friendship that makes my life pleasant and joyful

dis-The capacity of a wireless communication system can be increased considerably by usingmultiple transmit and receive antennas The high-data rate provided by such Multiple-input-multiple-output (MIMO) communication systems make them promising for next-generation wireless communication Among these MIMO techniques, space-time blockcoding (STBC) has attracted much research interests The orthogonal structure ofSTBC allows every symbol transmitted to be decoupled at the receiver using only linearprocessing Such a symbol-by-symbol receiver is simple yet efficient in implementation

to achieve the gain provided by both transmit and receive diversities

To coherently decode the STBC, ideally perfect channel state information (CSI)would be used at the receiver As the channel information is not readily available

at the receiver in practice, channel estimates are used to perform coherent detection.The optimum maximum likelihood detector with imperfect channel estimation is farmore computationally complicated than the optimum symbol-by-symbol detector whenperfect CSI is available In this dissertation, we propose a symbol-by-symbol channelestimation receiver for STBC systems, which is sub-optimal but computationally efficientfor implementation and can be applied to many channel models with their correspondingestimators In particular, we analyze the bit error probability (BEP) performance of thisreceiver when minimum mean-square-error estimates are available

We first derive the BEP performance of the receiver with maximum ratio combining.The BEP result is given in an exact closed-form expression, which shows the directdependence on the mean square error of the channel estimator and the signal-to-noiseratio An upper bound is derived to show the maximum diversity order achievable, which

is determined by the product of the numbers of transmit and receive antennas Wethen extend the work to a system with selection combining schemes, where the receiver

selects the received signal from one or several antennas with best quality according tothe channel estimates Exact closed-form BEP expressions are derived The resultsshow that the selection combining systems achieve the diversity gain provided by thetotal number of available receive antennas, but independent of the number of antennaschosen.

Transmit antenna selection (TAS) is a technique to exploit the transmit diversityother than space-time coding We propose a TAS/STBC system based on the channelestimation receiver structure Through a feedback link, the receiver informs the trans-mitter which antennas to be used for STBC transmission This TAS/STBC system has

a simple yet energy-saving structure, while exhibits the full diversity order provided bythe total number of transmit antennas An BEP upper bound is obtained in closed-form for the TAS/STBC systems Particularly, exact BEP expressions are derived forTAS/STBC systems with single receive antenna, which is important in down-link com-munication scenarios

The designs of orthogonal STBC so far known are limited Unitary space-timemodulation (USTM) treats the whole transmission block as one constellation, and thusprovides many more possible designs while maintaining the orthogonality of signals.However, there is no systematic method for optimal USTM constellation design Thus wepropose a systematic algorithm to search for sub-optimal differential unitary space-timemodulation The constellations generated by the proposed simple algorithm exhibitsbetter performance than the well-known cyclic codes

In summary, in this dissertation, space-time block coded communication systemswith imperfect channel estimation are extensively studied and BEP performances areobtained in closed-forms Improved algorithms for constellation search are also proposedfor differential unitary space-time modulation systems

Abstract i

1.1 Introduction to Wireless Communication Systems 1

1.2 A Literature Review of Space-Time Coding 2

1.2.1 Simulcast 3

1.2.2 BLAST 3

1.2.3 Space-Time Trellis Codes 4

1.2.4 Space-Time Block Codes 5

1.2.5 Unitary Space-Time Modulation 6

1.2.6 MIMO Applications in 3G Wireless Systems and Beyond 7

1.3 Research Objective 8

1.4 Structure of the Dissertation 9

1.5 Research Contributions 9

2 MIMO Communication Systems in Wireless Fading Channels 11

2.1 Capacity of MIMO Systems 11

2.1.1 MIMO Communication System 11

2.1.2 Capacity Analysis of MIMO Communication System 13

2.2 Mobile Radio Channels and MMSE Channel Estimation 17

2.2.1 Rayleigh Fading Channel with Butterworth power spectrum density 18 2.2.2 Kalman Filtering for State-Space Channel Model 23

2.2.3 Rayleigh Fading Channel with Jakes’ PSD 26

2.2.4 Wiener Filtering for Jakes’ Model 30

2.3 Phase-Shift Keying Modulation 31

2.4 Summary 32

3 BEP Performance Analysis of Orthogonal Space-Time Block Codes 33 3.1 Introduction 34

3.2 Receiver Structure for Orthogonal STBC 37

3.2.1 Definition of Orthogonal STBC 37

3.2.2 Transmitter Structure 38

3.2.3 Receiver Structure 38

3.2.4 Channel Estimator Structure 39

3.2.5 Optimum Receiver Structure 43

3.2.6 A Symbol-by-Symbol Receiver Structure 44

3.3 BEP Performance Analysis for OSTBC Systems 46

3.4 Numerical Results and Discussion 53

3.5 Summary 63

4 STBC Communication System with Receive Antenna Selection 64 4.1 Introduction 65

4.2 System Model and Receiver Structure 66

4.3 Performance Analysis of STBC with Selection Combining 69

4.3.1 Single selection combining 72

4.3.2 Generalized Selection Combining 76

4.4 Numerical Results and Discussion 78

4.4.1 Single Selection Combining 79

4.4.2 Generalized selection combining 84

4.5 Summary 86

5 STBC Communication System with Transmit Antenna Selection 88 5.1 Introduction 88

5.2 System Model 89

5.3 Performance Analysis of STBC with TAS 92

5.3.1 An Upper Bound for BEP 93

5.3.2 Exact BEP Analysis for TAS Systems 94

5.4 Numerical Results and Discussion 96

5.5 Summary 105

6 Constellation Design for Unitary Space-Time Modulation 106 6.1 Unitary Space-Time Modulation 106

6.1.1 Constellations that Achieve Capacity 106

6.1.2 Unitary Space-Time Modulation 109

6.1.3 Differential Unitary Space-Time Modulation 110

6.1.4 Constellation Design Criteria for DUSTM 114

6.1.5 A Revisit of Cyclic Designs 118

6.2 Constellation Design for Unitary Space-Time Modulation 119

6.2.1 DUSTM Constellation Designs Based on Rotation Matrices (Scheme I) 120

6.2.2 DUSTM Constellation Designs Based on Full-Rotation Matrices

(Scheme II) 1256.3 Numerical Results and Discussion 1316.4 Summary 136

7 Conclusions and Proposals for Future Research 1377.1 Conclusions 1377.2 Proposals for Future Research 139

1.1 Delay Diversity and Trellis Space-Time Code 5

2.1 MIMO System Model 12

2.2 Communication channel model 18

2.3 Markov signal model for Kalman filter 19

2.4 Simulated p.d.f of 1BTW channel model 20

2.5 Simulated correlation functions of 1BTW channel model 21

2.6 Simulated p.d.f of 3BTW channel model 23

2.7 Simulated correlation functions of 3BTW channel model 24

2.8 Kalman Filter Structure 24

2.9 Simulated p.d.f of Jakes’ channel model 26

2.10 Simulated correlation functions of Jakes’ channel model 27

2.11 Channel samples of size one thousand for different models 29

2.12 Linear Wiener Filter Model 31

2.13 Constellation maps of PSK signaling 32

3.1 Decision feedback channel estimation STBC system 41

3.2 PSAM channel estimation STBC system 41

3.3 PSAM frame structure 42 3.4 Theoretical BEP performance of Alamouti’s STBC under BTW channel 54 3.5 Theoretical BEP performance of Alamouti’s STBC under 1BTW channel 55

LIST OF FIGURES

3.6 Theoretical BEP floor under BTW channel 56

3.7 Theoretical BEP performance with multiple transmit antennas 58

3.8 Theoretical comparison between full- and half- rate STBC’s 59

3.9 Theoretical bounds of BEP performance for different STBC’s 60

3.10 BEP of BPSK with Alamouti’s STBC with one receive antenna 61

3.11 BEP performance of 4 × 4 rate-3/4 STBC with 3BTW 62

3.12 BEP performance of 4 × 4 rate-3/4 STBC with PSAM 63

4.1 System model of STBC with selection combining 69

4.2 BEP Performance of 1-Tx system with single selection combining 79

4.3 Performance comparison between MRC and SSC systems 80

4.4 Performances QPSK and 8PSK modulation with SSC and Alamouti’s STBC 81 4.5 Performance comparison among different STBC’s with SSC 82

4.6 Performance comparison of different STBC’s against channel fade rate 83

4.7 Performance of GSC with 1-Tx and 4-Rx 84

4.8 Performance of Alamouti’s STBC with dual selection combining 85

4.9 Mean output of the estimated SNR with single and dual selection combining 86 5.1 System model of STBC with TAS 90

5.2 Performance of Alamouti’s STBC with transmit antenna selection 97

5.3 Performance of the 4 × 4, rate 3/4 STBC with transmit antenna selection 98 5.4 Performance comparison among different STBC’s with MX = 4 99

5.5 Performance comparison among different STBC’s with MX = 8 100

5.6 Performance comparison between TAS and STBC with MX = 2 101

5.7 Theoretical and simulation performances of Alamouti’s STBC with MX = 4 102

5.8 Performances comparison of TAS and STBC with MX = 2 103

5.9 Performances comparison of TAS and STBC with MX = 4 104

6.1 Diversity product sample ζ0l0 when MT = 4, L = 16 113

6.2 Diversity product function distribution with constellation size L = 16 129

6.3 Demonstration of algorithm complexities 130

6.4 SEP of DUSTM with MT = 2 and L = 5, 7, 9 132

6.5 SEP of DUSTM with MT = 2 and L = 8, 16, 32, 64 132

6.6 SEP of DUSTM with MT = 2 under fast fading 133

6.7 SEP of DUSTM with MT = 4, L = 4, 32, 64 133

6.8 SEP of DUSTM with MT = 8, L = 8, 32, 64 134

6.9 SEP of DUSTM with L = 64 under fast fading 134

7.1 Examples of relay diversity 141

7.2 Selection relay diversity 142

List of Tables

3.1 Parameters list for exact BEP evaluation 533.2 Parameters list for lower and upper bound of BEP evaluation 536.1 Diversity products of different constellation design schemes 1246.2 Comparison of Diversity products of Scheme II against cyclic codes 1276.3 Diversity product and diversity sum of the proposed constellation 1286.4 Run-time comparison of algorithms 130

ADF actual decision feedback

AWGN additive white Gaussian noise

BEP bit error probability

BLAST Bell-lab Layered Architecture of

Space-Time

COD complex orthogonal designs

CSI channel state information

DD differential detection

DF decision-feedback

DSC dual selection combining

DUSTM differential unitary space-time

modula-tion

EGC equal gain combining

EM expectation-maximization

GSC generalized selection combing

i.i.d independent, identically distributed

IDF ideal decision feedback

MMSE minimum mean squared error

MRC maximum ratio combining

MSE mean square error

p.d.f probability density function PEP pair-wise error probability PSAM pilot-symbol assisted modulation PSD power spectrum density

PSK phase-shift keying r.v random variable

Rx receive antenna SBS symbol-by-symbol SEP symbol error probability SIMO single-input-multi-output SNR signal-to-noise ratio SSC single selection combining STBC space-time block codes STTC space-time trellis codes TAS transmit antenna selection TCM trellis coded modulation

Tx transmit antenna USTM unitary space-time modulation

WF Wiener filter WLAN wireless local-area network

Chapter 1

Introduction

The ability to communicate with people on the move has evolved remarkably eversince 1897, when Guglielmo Marconi first demonstrated continuous contacts with shipssailing the English Channel using a radio More recently, the technical breakthroughs indigital and radio frequency circuit fabrication, new large-scaled circuit integration andother miniaturization technologies have made the portable radio equipment smaller,cheaper by orders of magnitude for the past several decades, and will continue at aneven greater pace for the coming decade

Sys-tems

More than 20 years have passed since the first-generation mobile communicationservices using analog technology started in the early 1980s From the early 1990s, digitalcellular and cordless systems (e.g PDC/GSM/IS54 and IS95) have been introducedaround the world as the second-generation (2G) mobile communication systems capable

of voice and short message communications The 2G services have been integratedinto our everyday life and society extensively after explosive growth for more than ten

years Meanwhile, research and standardization have been carried out toward the generation (3G) mobile communication systems for the past decade, which is capable

third-of mobile multimedia services and international seamless roaming Telecommunicationcompanies worldwide are now beginning to deploy 3G systems for commercial serviceand we will soon be in the era of 3G As for researchers and engineers, they have alreadyput their sight to a highly reliable and higher capacity wireless digital system, which is to

be called as the fourth-generation (4G) mobile radio communication systems The generation requires high speed reliable wireless systems for multimedia communicationsservices, including voice, data, and image

next-The tremendous growth in demand for higher data rates is now out of the range ofcurrent radio technology Given a limited radio spectrum, the only way to support highdata rates is to develop new spectrally efficient radio communication techniques

Wireless transmission under fading channel suffers from attenuation due to structive addition of multipaths in the propagation media and due to the reflec-tions,scatterings, interference from other users, etc Severe attenuation makes it im-possible for the receiver to determine the transmitted signal unless some less-attenuatedreplica of the transmitted signal is provided to the receiver This resource is calleddiversity and it is the single most important contributor to reliable wireless communi-cations Examples of diversity techniques are, but not restricted to, temporal diversity,frequency diversity, and antenna diversity Conventionally, to exploit the receive an-tenna diversity, multiple antennas are deployed at the receiver side to increase the linkcapacity Recently, researchers have found ways to deploy multiple antennas at thetransmit side to further increase the communication capacity Thus a communicationsystem with multiple transmit and multiple receive antennas is formed, and we call it a

de-1.2 A LITERATURE REVIEW OF SPACE-TIME CODING

multiple-input-multiple-output (MIMO) communication system A brief historic review

of MIMO systems is given as following

1.2.1 Simulcast

The concept of MIMO system can be traced back to 1987, when Winters proposedtwo basic communication systems in [1]: communication between multiple mobiles and abase station with multiple antennas, and communication between two mobiles each withmultiple antenna This is the first paper that discusses the use of multiple antennas atboth ends of the radio link and gives the capacity expression in terms of the eigenvalues

of the channel matrix In [2] and [3], the authors considered a communication networkwhere several adjacent base station simultaneously transmit the same message Later,and independently, a similar scheme was suggested by Seshadri and Winters for a sin-gle base station in which copies of the same symbol are transmitted through multipleantennas at different times [4], hence creating an artificial multipath distortion Then

a maximum likelihood sequence estimator or a minimum mean squared error (MMSE)equalizer is used to resolve multipath distortion and obtain diversity gain

Subsequently, Foschini presented the analytical basis of MIMO systems in [5, 6],where he proposed key expressions for the enhanced capacity of MIMO systems Refer-ence [5] is the first paper in which Bell Lab proposed BLAST (Bell-lab Layered Archi-tecture of Space-Time) as communication architecture for the transmission of high datarates using multiple antennas at the transmitter and receiver In the proposed BLASTsystem the data stream is divided into blocks which are distributed among the transmitantennas In vertical BLAST sequential data blocks are distributed among consecutiveantenna elements, whereas in diagonal BLAST, they are circularly rotated among theantenna elements The BLAST signal processing algorithms used at the receiver are

the heart of the technique At the bank of receiving antennas, high-speed signal sors look at the signals from all the receive antennas simultaneously, first extracting thestrongest substream and then proceeding with the remaining weaker signals, which areeasier to recover once the stronger signals have been removed as a source of interference.Again, the ability to separate the substreams depends on the slight differences in theway the different substreams propagate through the environment.

proces-Under the widely used theoretical assumption of independent Rayleigh scattering,the theoretical capacity of the BLAST architecture grows roughly linearly with thenumber of antennas, even when the total transmitted power is held constant Thelaboratory prototype [7] has already demonstrated spectral efficiencies of 20 - 40 bits persecond per Hertz of bandwidth, numbers which are simply unattainable using standardtechniques

1.2.3 Space-Time Trellis Codes

Although the first attempt to jointly encode multiple transmit antennas was sented in [4], the key development of the space-time coding concept was originally re-vealed in [8] in the form of trellis codes Somehow, space-time trellis codes (STTC)can be viewed as an improvement of the delay diversity scheme The example trellisdiagram of delay diversity is shown below in Figure 1.1 By simply swapping the oddrow of the delay-diversity trellis diagram, 2.5-dB coding gain can be achieved in (b),which is a typical STTC Note that the STTC is also a delay scheme except the delayedPSK symbol is π-shifted on the constellation plane if it is an odd symbol, and kept thesame if even symbol

pre-The STTC requires a multidimensional Viterbi algorithm at the receiver for ing It was shown in [8, 9] that the STTC provides a diversity gain equal to the number

decod-of transmit antennas, and a coding gain which depends on the complexity decod-of the code,i.e., number of states in the trellis, without any loss in the bandwidth efficiency Still

1.2 A LITERATURE REVIEW OF SPACE-TIME CODING

Fig 1.1: Delay Diversity and Trellis Space-Time Code (Figure partially taken from [8])

the gain of STTC is achieved at the expense of a complex receiver Since the debut ofSTTC in [8], there has been extensive research aiming at improving the performance ofthe original STTC designs Numerous works have been proposed for new code construc-tion and designs of STTC systems, e.g., [10–14] However, only marginal gains over theoriginal scheme by Tarokh et al were obtained in most cases

1.2.4 Space-Time Block Codes

The receiver complexity of STTC increases exponentially with the dimensions ofcode, trellis, etc., thus making the receiver structure quite complex in implementation.The popularity of space-time coding really took off with the discovery of the so-calledspace-time block codes (STBC) In [15], Alamouti presented a perfectly beautiful codethat exploits the transmit diversity with two transmit antennas The orthogonal con-struction of the code allows simple linear processing at the receiver, in contrast to themulti-dimensional Viterbi decoder at the STTC receiver Later, Tarokh et al gener-alized this scheme for an arbitrary number of transmit antennas[16, 17] While STBC

provides the same diversity gain as STTC, it gives none or minimal coding gain.

The coherent detection in both [15] and [17] requires perfect channel state tion (CSI) at the receiver In [18] and [19], differential STBC schemes were presented,respectively, for Alamouti’s code and generalized STBC with an arbitrary number oftransmit antennas The authors use some mapping skills to determine the next block

informa-to be sent Similar informa-topics were also addressed in [20–22] More complicated differentialdesigns can also be found in [23, 24] to combat the fading

More recently, a new scheme called unitary space-time modulation (USTM) [25]was proposed to achieve channel capacity The key idea of the USTM is that the wholetransmitting matrix is treated as one constellation signal By constraining the signalmatrix to be unitary, it is proved that the USTM is still capacity-achieving Moreover,there are more available designs compared to the limited designs of STBC, since the entry

of USTM signal matrix is no longer restricted to the combination of certain symbols from

a given constellation set In [25] and [26], it is pointed out that the ultimate capacity of

a multiple-antenna wireless link is determined by the number of symbol periods betweenfades The diversity gain achievable is constrained by the coherent symbol periods Forexample, in the extreme case where the channel fluctuates every symbol period, onlyone transmitter antenna can be usefully employed Theoretically speaking, one couldincrease the capacity indefinitely by employing a greater number of transmit antennas,but the capacity appears to increase only logarithmically in this number - not a veryeffective way to boost capacity So, actually, there is no point in making the number oftransmitter antennas greater than the length of the coherence interval

When the coherence interval becomes large compared with the number of ter antennas, the normalized capacity approaches the capacity obtained as if the receiverknew the propagation coefficients The magnitudes of the time-orthogonal signal vectors

transmit-1.2 A LITERATURE REVIEW OF SPACE-TIME CODING

become constants that are equal for all transmitter antennas In this regime, all of thesignaling information is contained in the directions of the random orthogonal vectors,the receiver learns the propagation coefficients, and the channel becomes similar to theclassical Gaussian channel

1.2.6 MIMO Applications in 3G Wireless Systems and Beyond

The 3G mobile communications standards are expected to provide a wide range

of bearer services, spanning from voice to high-rate data services, supporting rates of

at least 144 kb/s in vehicular, 384 kb/s in outdoor-to-indoor and 2 Mb/s in indoor aswell as pico-cellular applications In work beyond 3G the target is to achieve data rates

in the order of 1Gbps for low-mobility solutions, and 100 Mbps for full coverage andmobility

Some techniques like turbo coding have brought the utilization of a single link veryclose to Shannon limits of channel capacity The next step is the creation of multiplelinks between a terminal and a base station,which is fulfilled by MIMO systems Sofar there is little commercial implementation of MIMO in cellular systems and deployed3G systems The existing MIMO applications include the Lucent’s BLAST chip, which

is demonstrated to be capable of high data rate transmissions Recently, the generation partnership project has standardized the MIMO models in IEEE 802.16.Also in the standard IEEE 802.11n for wireless local-area network (WLAN) , MIMOtechniques have been adopted to boost the data rate Multiple commercialized modelswith MIMO techniques have recently been released [27], which demonstrate impressiveperformances gains against the existing products With the potential communicationcapacity provided by the multiple links, it is predictable that MIMO systems will beincorporated into wireless communications of most kinds: cellular, WLAN, or evensatellite in the near future

third-1.3 Research Objective

As addressed above, the MIMO system is an attractive solution for the generation wireless communication In our research, we have concentrated on theperformance analysis of STBC systems and differential unitary space-time modulation(DUSTM)

next-In STBC system designs,it is assumed that the receiver knows perfectly the CSI forcoherent detection Although differential schemes have been proposed which do not needCSI, they actually require the channel coherence interval to be long enough for efficientdetection When the channel fluctuates faster, the performance of differential schemesdegrades considerably This makes an STBC system that is incorporated with channelestimation more preferable in practice The objective of our research is to develop such

a receiver with channel estimation and analyze its performance under fading channels.Space-time coding provides us with transmit diversity additional to those diversitiesconventionally used In receive antenna diversity, we have several combining schemes

to utilize those received signals undergoing more-or-less independent fading, e.g., equalgain combining (EGC) , maximum ratio combining (MRC) , selection combining, etc.Those schemes can all be independently adopted at the receiver for MIMO systems.Thus, it also aroused our interest in what the performance will be if we introduce thesereceive diversity combining techniques together with the transmit diversity provided

by the space-time coding Also, for a communication system with multiple transmitantennas, if the transmitter knows the channel fading, it can choose the best one orseveral antennas to transmit The design and performance of such an adaptive transmitsystem is also within our research interests

Furthermore, finding good constellation sets is always of interest for MIMO systems.This problem is still open since so far there is no systematic optimum solution We alsoput our effort into this approach to find simple yet efficient constellation designs

1.4 STRUCTURE OF THE DISSERTATION

In the next chapter, we present some basic background on MIMO systems and thechannel model adopted in this dissertation

In Chapter 3, we propose a symbol-by-symbol channel-estimation receiver structurefor STBC systems Based on the receiver structure, we analyze the performance of thereceiver with imperfect channel estimation

In Chapter 4, we concentrate on the receiver structure developed in Chapter 3together with selection combining Bit error probability (BEP) performance analysis iscarried out based on the order statistics of estimated SNR

We further extend the work by feeding back the channel estimation information tothe transmitter to optimize the performance We present an adaptive transmit antennaselection system System structure and performance analysis are presented

In Chapter 5, two new methods for DUSTM constellation design are proposed Thealgorithms are described in detail The new methods provide better performance thanthe known cyclic codes, yet with limited increase in computational complexity

We develop a receiver structure for STBC system with imperfect channel estimation

in Chapter 3 As the optimal maximum-likelihood receiver is rather computationallycomplex, we use a symbol-by-symbol receiver for its simplicity Based on this symbol-by-symbol receiver structure, performance analysis is carried out to predict its BEP withphase-shift keying modulations A closed-form BEP expression is obtained for thoseSTBC’s where energy is uniformly distributed along time For those STBC’s whereenergy is not uniform along time, upper and lower bounds are obtained to predict theperformance These two bounds are in most cases so close to each other that they providegood approximation to the exact BEP Simulations conducted validate our theoretical

Based on the results obtained in Chapter 3, we further extend our work to nel estimation STBC systems with receive antenna selection combining and transmitantenna selection in Chapter 4 and 5, respectively In both receive antenna selectionand transmit antenna selection schemes, the choice of the transmit/receive antennas arebased on the channel estimates, i.e., it is a channel-estimation based system, so that noexpensive and complex signal-to-noise ratio (SNR) evaluation is needed at the receiver,which reduces the complexity of the receiver to a large extent Based on the systemstructures, BEP performances are derived and presented in closed-form expressions

chan-We improved the cyclic code presented in [28, 29] for DUSTM To utilize the time diversity more than the cyclic code does, we introduce a rotation matrix in codeconstruction The proposed constellations are in quasi-diagonal matrix forms Detaileddiversity product calculations are analyzed to simplify the search process The finalalgorithm improves the diversity product significantly compared to the cyclic codes,with limited increase or even reduced computational complexity

We consider a MIMO system with MT transmit and NR receive antennas as shown

in Figure 2.1

The transmitted signal at time p is represented by an 1 × MT row vector S =[sp1, sp2, , spMT] The total transmitted power is constrained to E0, regardless of the

Fig 2.1: Wireless link with M T transmitter and N R receiver antennas Every receiver antenna is

con-nected to every transmitter antenna through an independent, random, unknown propagation coefficient having Rayleigh distributed magnitude and uniformly distributed phase Normal- ization ensures that the total expected transmitted power is independent of M T for a fixed ρ

number of transmit antennas MT This power constraint gives

be considered as flat

The channel is described by an MT × NR complex matrix, denoted by H, whoseelement hil represents the propagation coefficient between the i-th transmit antenna andthe l-th receive antenna For normalization purposes we assume that the received powerfor each of the NR receive antennas is equal to the total transmitted power, i.e., E0

2.1 CAPACITY OF MIMO SYSTEMS

Thus we obtain the normalization constraint for the elements of H, in a channel withfixed coefficients, as

We assume that the channel matrix is known to the receiver when using a method such

as transmitting training preamble On the other hand, in most situations we assumethat the channel parameters are not known at the transmitter

At the receiver, the additive noise is described by an 1 × NR row matrix N =[np1, np2, , npNR], whose components are statistically independent, complex, zero-meanGaussian variables The receive antennas have identical noise powers of N0

The received signal is represented by an 1 × NR row matrix, denoted by R =[rp1, rp2, , rpNR], where each complex component refers to a receive antenna

The average SNR at each receiver branch is given by

The channel capacity is defined as the maximum possible transmission rate suchthat the probability of error is arbitrarily small The well-known Shanon capacity is

given by

C = W log2(1 + ρ), (2.6)where W is the bandwidth of the communication channel and ρ is the SNR In thesystem mentioned above, according to the singular value decomposition, the channelmatrix H can be written as

where U and V are MT × MT and NR× NR unitary matrices, respectively, and D is an

MT × NR non-negative diagonal matrix given by

R0 =√

ρDS0+ N0, (2.11)where R0 = RU , S0 = SV and N0 = N U

Since τ = rank(H) = rank(H†H), for the MT × NR matrix H, the rank τ is at

2.1 CAPACITY OF MIMO SYSTEMS

most m = min (MT, NR), which means that at most m of its singular values are non-zero

By substituting the entries σi in (2.11) we get for the received components

r0l= σls0pl+ n0l, l = 1, 2, , τ

r0l= n0l, l = τ + 1, τ + 2, , NR

(2.12) indicates that the received components r0l, l = τ + 1, τ + 2, , NR, do not depend

on the transmitted signal, i.e., the channel gain is zero On the other hand, receivedcomponents rpl0 , for l = 1, 2, , τ depend only on the transmitted component s0pl Thusthe equivalent MIMO channel from (2.11) can be considered as consisting of τ uncoupledparallel channels For example, if MT > NR, as the rank of H cannot be higher than NR,(2.12) shows that there will be at most NRnon-zero gain sub-channels in the equivalentMIMO channel On the other hand if NR > MT, there will at most MT non-zero gainsub-channels in the equivalent MIMO channel

Note that in the above models the sub-channels are uncoupled and thus their pacities are summed up Assuming that transmit power from each antenna is identical,from (2.1), we can estimate the overall channel capacity as

Ei =



µ − E0/ρ

σ2 i

+

i = 1, 2, , τ, (2.15)

where a+ denotes max(a, 0) and µ is determined so that

2 i

This capacity corresponds to that of linear maximum ratio combining at the receiver

In the case when the channel matrix elements are equal and normalized as follows:

|h1|2 = |h2|2 = |hMT|2 = 1, (2.20)then the capacity becomes

C = W log2(1 + ρ) (2.21)This expression applies to the case when the transmitter does not know the channel.For coordinated transmissions, when the transmitter knows the channel, we can applythe capacity formula from (2.17) As the rank of the channel matrix is one, there is onlyone term in the sum in (2.17) and only one non-zero eigenvalue given by

2.2 MOBILE RADIO CHANNELS AND MMSE CHANNEL ESTIMATION

And from the normalization condition, we have

For MT = 8 and SNR of 20dB, the capacity is 9.646bps/Hz

Es-timation

The communication channel is the physical medium that connects the transmitterand the receiver It can be a pair of wires or an optical fiber for wired communication Inwireless communication environment, the channel is the free space between the transmitand the receive antennas The presence of reflecting objects and scatterers in the spacecreates a constantly changing environment that dissipates the signal energy in amplitude,phase, and time These effects result in multiple versions of the transmitted signal thatarrive at the receiving antenna, displaced with respect to one another in time and spatialorientation The random phases and amplitudes of the different multipath componentscause fluctuations in strength of the received signal There are many channel models inthe literature Here, in this dissertation, we consider the non frequency-selective Rayleighchannel models, where the received signal is a summation of many reflected signals andthe signal with maximum delay does not exceed the symbol duration Assume that themultipaths are independent and statistically identical, and the number of multipaths islarge enough, the fading gain can then be modeled as a complex symmetric Gaussianrandom variable The absolute value of the complex Gaussian gain follows the Rayleighdistribution This non frequency-selective slow Rayleigh fading channel is the most

Fig 2.2: Communication channel model

widely accepted channel model for narrowband transmission systems

Fig.2.2 depicts the baseband channel model we used in this dissertation The tiplicative channel gain is introduced by the medium while the additive white Gaussiannoise (AWGN) arises from the electronic circuitry in the receiver

mul-At the receiver, perfect sampling is assumed and thus the multiplicative channelgain is assumed to be piecewise constant for a symbol duration The model in Fig.2.2 can be expressed in a discrete-time representation for the m -th symbol duration[mTs, (m + 1)Ts] where Ts is the symbol duration time

or other channels where the fading process is exponentially correlated Here, the in-phasecomponent xc(m) = Re [h(m)] and the quadrature-phase component xs(m) = Im [h(m)]

of each fading process is the output of a state-space model, i.e., xc(m) or xs(m) =Bx(m), where the state vector x(m) evolves according to a model

x(m + 1) = F x(m) + G w(m) (2.26)

2.2 MOBILE RADIO CHANNELS AND MMSE CHANNEL ESTIMATION

Fig 2.3: Markov signal model for Kalman filter

of appropriate dimension Substituting (2.26) into the communication channel model(2.25), one get the received signal model depicted in Fig 2.3, where {w(m)}∞m=0 and{v(m)}∞m=0 are zero mean, independent Gaussian processes with covariance matricesgiven by

E[w(m)wT(m0)] = W δmm0 (2.27a)E[v(m)vT(m0)] = V δmm0 (2.27b)E[w(m)vT(m0)] = 0 (2.27c)After x(m) has been generated according to 2.26, it is sent to a multiplicative channelwith gain ˜HT(m) ; a sample z(m) is obtained by further disturbing the so generatedy(m) with an additive noise v(m) z(m) is the noisy sample available to the filter torecover x(m), which will be discussed in the next section We consider here in particularthe case of a first-order Butterworth (1BTW) and a third-order Butterworth (3BTW)model for the channel

Fig 2.4: Theoretical and simulated PDF’s of the real part of the first-order Butterworth channel model

with ωdTs= 0.01 and σ 2 = 0.25 The simulated PDF is obtained by averaging 1000 repeated trials

2.2.1.1 First-Order Butterworth Channel Model

The first-order Butterworth fading process has the following power spectrum density(PSD) function:

S(ω) = 2σ

2/ωd

1 + (ω/ωd)2, (2.28)where ωd is the 3dB radian frequency of the Butterworth power spectrum The state-space realization for the Rayleigh fading channel with first-order Butterworth PSD isobtained by transforming (2.28) into time domain and solving its first order differen-tial equation The processes {xc(m)}∞m=0 and {xs(m)}∞m=0 are each given by a one-dimensional version of (2.26) with

F = exp [−ωdTs] , G = 1, B = 1, and W = σ2(1 − e−2ωd T s), (2.29)where Ts is the interval between discrete time points

The output probability density function (p.d.f.) of the simulator given in (2.26) and

2.2 MOBILE RADIO CHANNELS AND MMSE CHANNEL ESTIMATION

Fig 2.5: (a)Theoretical and simulated autocorrelation function of the real part;

(b) Theoretical and simulated crosscorrelation function between the real and imaginary part ;

of the first-order Butterworth channel model with ω d T s = 0.01 and σ 2 = 0.25 The simulated results are obtained by averaging 1000 repeated trials

(2.29) is plotted in Fig 2.4 Compared to the theoretical one, the output samples have

a perfect Gaussian distribution as designed In this dissertation, we are more interested

in the channel autocorrelation function than the PSD, since the autocorrelation functionwould be used to evaluate the mean square error(MSE) of channel estimator According

to the 1BTW’s PSD function in (2.28), the autocorrelation function is obtained byinverse Fourier transforming

Rxcxc(∆m) = Rxsxs(∆m) = σ2e−ωd T s |∆m|

(2.30)

and the cross-correlation Rxsxc(∆m) or Rxcxs(∆m)between the real and imaginary partsshould be zero as they are zero-mean independent processes In Fig 2.5, we plot thesimulated correlation function Rxcxc(∆m) and Rxcxs(∆m) for example It is shown thatthey match the theoretical prediction quite well.

2.2.1.2 Third-Order Butterworth Channel Model

In the 3BTW model, the quadrature components of the continuous-time fadingprocess each have a power density spectrum

S(ω) = 3σ

2

ωd[1 + (ω/ωd)6]. (2.31)The processes {xc(m)}∞k=0 and {xs(m)}∞k=0 can each be generated using a three-dimensional version of the model (2.26) with

B = h 1 0 0 i, W = 3σ2ωdT (2.32)

Note that in achieving the Markov model given in (2.26) and (2.32), approximation

ωdTs << 1 must been taken Therefore, the simulation model is only fine for slowfading The p.d.f of 3BTW with ωdTs = 0.001 is plotted in 2.6 The simulated resultstill matches theoretical one quite well It is reported through simulation that thesimulation output has a 5% greater variance than the desired one As the ωdTs furtherincrease, the deviation from theoretical becomes more and more obvious, and thus makesthe simulator deviate from theory under fast fading situations So the usage of 3BTW

is refrained to slow fading scenarios The 3BTW PSD in (2.31) has an autocorrelation

2.2 MOBILE RADIO CHANNELS AND MMSE CHANNEL ESTIMATION

Fig 2.6: Theoretical and simulated PDF’s of the real part of the third-order Butterworth channel model

with ωdTs= 0.001 and σ 2 = 0.25 The simulated PDF is obtained by averaging 1000 repeated trials

(2.33)

The theoretical correlation functions together with simulated ones are illustrated

in Fig 2.7 Perfect matches are observed, which validates that the 3BTW simulator isstill quite good under slow fading

2.2.2 Kalman Filtering for State-Space Channel Model

When a state-space channel model is available, the Kalman filter (KF) is the mum channel estimator The KF is more suitable for the decision-feedback (DF) channelestimation scheme since it can operate recursively in time as symbol decisions are made.According to the state-space signal model in Fig 2.3, the correspondent Kalman fil-ter structure is shown in Fig 2.8 The principle of KF is described by the following

opti-Fig 2.7: (a)Theoretical and simulated autocorrelation function of the real part;

(b) Theoretical and simulated crosscorrelation function between the real and imaginary part ;

of the third-order Butterworth channel model with ωdTs= 0.001 and σ 2 = 0.25 The simulated results are obtained by averaging 1000 repeated trials

Fig 2.8: Kalman Filter Structure

2.2 MOBILE RADIO CHANNELS AND MMSE CHANNEL ESTIMATION

equations ([32], Chap 3, eq.(1.9) and eq.(1.12)

P (m) = H˜T(m)Σ(m|m − 1) ˜H(m) + V (m) (2.34)K(m) = F (m)Σ(m|m − 1) ˜H(m)P−1(m) (2.35)ˆ

x(m|m) = x(m|m − 1) + K(m)[z(m) − ˜ˆ HT(m)ˆx(m|m − 1)] (2.36)Σ(m|m) = Σ(m|m − 1) − Σ(m|m − 1)K(m) ˜H(m) (2.37)ˆ

Σ(m + 1|m) = F (m)Σ(m|m)FT(m) + G(m)W (m)GT(m) (2.39)where K(m) is the Kalman gain; ˆx(m+1|m) and ˆx(m|m) are the predicted and updatedstate vector; and Σ(m+1|m) and Σ(m|m) are the predicted and updated error covariancematrices With initialized value ˆx(0| − 1) and Σ(0| − 1), the KF recursively computesequations (2.34) through (2.39) and predict the MMSE estimate ˆx(m+1|m) for x(m+1)

As the exact initial value of ˆx(0| − 1) and Σ(0| − 1) are unlikely known to the KF, the

KF needs several periods to establish a track-on state before it can predict reliably Thisinitialization phase is done in communication system by sending known preambles tothe receiver before data transmission

Note that (2.34),(2.35),(2.37), and (2.39) are independent of the observation z(m),they can be calculated off-line These equations are also known as Riccati equations

As Σ represents the MSE of the estimates, by solving those four equations recursively, asteady-state value of the MSE can be obtained Especially for the one-dimensional case,the MSE has a closed-form expression as

Σ∞= V F

2+ G2W ˜H2− V +

q(V + V F2 + G2W ˜H2)2− 4V2F2

For KF more than one-dimensional, the MSE must be obtained by repeatedly computingthe Riccati equations until a steady-state is reached In Matlab, there is a Riccati()function in control toolbox to give readily the solution of Riccati equations

Fig 2.9: Theoretical and simulated PDF’s of the real part of the Jakes’ simulator ωdTs = 0.1 and

σ 2 = 0.25 The simulated PDF is obtained by averaging 1000 repeated trials

2.2.3 Rayleigh Fading Channel with Jakes’ PSD

The Jakes spectrum [33, 34] is commonly used to model the fading process for theland-mobile cellular channel This Jakes power density spectrum is defined as

Over the last three decades, there are quite a lot of different approaches to thesimulation model of Jakes’ model [34] The most well-known mathematical referencemodel by Clarke [33] and its simplified simulation model by Jakes [34] have been widelyaccepted for Rayleigh fading channels for more than thirty years However, the Jakes’simulator is a deterministic model, and the result is questionable when generating mul-

2.2 MOBILE RADIO CHANNELS AND MMSE CHANNEL ESTIMATION

Fig 2.10: (a)Autocorrelation function of the real part;

(b)Autocorrelation function of the imaginary part;

(c) Cross-correlation function between the real and imaginary part ;

(d)Real part of the autocorrelation function of the entire complex process;

(e)Imaginary part of the autocorrelation function of the entire complex process;

(f)Autocorrelation function of the output envelope;

of the Jakes’ channel model with ω d T s = 0.01 and σ2 = 0.25 The simulated results are obtained by averaging 1000 repeated trials

tiple uncorrelated fading process for frequency selective fading channels and MIMOchannels Therefore, modifications of Jakes’ simulator have been proposed [35–38] Re-cently in [39], it was pointed out that the Jakes’ model is wide-sense nonstationarywhen averaged across the physical ensemble of fading channels An improved simulator